Problem 33 - Calculus and its applications

Question

Lottery winnings. Suppose you win \(\$ 5,000,000\) (after taxes) in a lottery. Instead of a one-time payment, you accept a structured plan of annual payments over 20 yr. Under this plan, you receive yearly payments \(W\) that assume an annual interest rate of \(5 \%\) and that will accrue to \(\$ 5,000,000\) over \(20 \mathrm{yr}\). Find the annual payment \(\mathrm{W}\) you receive under this plan.

Step-by-Step Solution

Answer

The annual payment is approximately $401,731.

Step 1: Understanding the Present Value Formula

To solve this problem, we will use the present value of an annuity formula. This formula helps us find the annual payment when we know the total value at the end of a certain period, using a particular interest rate. The present value of an annuity formula is given by:\[ PV = W \times \frac{{1 - (1 + r)^{-n}}}{r} \]where \(PV\) is the present value, \(W\) is the annual payment, \(r\) is the interest rate, and \(n\) is the number of years.

Step 2: Setting Up the Equation

In this problem, \(PV\) is the total winnings of \(\$ 5,000,000\), \(r\) is \(5\%\) or \(0.05\), and \(n\) is \(20\) years. Plug these values into the formula:\[ 5,000,000 = W \times \frac{{1 - (1 + 0.05)^{-20}}}{0.05} \]

Step 3: Simplifying the Annuity Formula

Calculate the term \((1 + 0.05)^{-20}\) first:\[(1 + 0.05)^{-20} = (1.05)^{-20} \approx 0.37689\]Then, compute the entire fraction of the formula:\[\frac{{1 - 0.37689}}{0.05} = \frac{0.62311}{0.05} = 12.4622\]

Step 4: Solving for the Annual Payment W

Use the simplified equation from Step 3 to find \(W\):\[5,000,000 = W \times 12.4622\]Now, solve for \(W\) by dividing both sides by \(12.4622\):\[W = \frac{5,000,000}{12.4622} \approx 401,730.95\]

Step 5: Conclusion

So, the annual payment \(W\) you will receive each year under this plan is approximately \(\$ 401,731\).

Key Concepts

Annual Payment Calculation

When you're trying to calculate the annual payment in a scenario where a total amount is distributed over many years, such as with lottery winnings, understanding the present value of an annuity formula is essential. This formula is useful because it factors in how money today is worth more than the same amount in the future due to potential earning capacity, illustrated through an interest rate.
\[ PV = W \times \frac{{1 - (1 + r)^{-n}}}{r} \]
Here's what you need to do:

• Identify the total future value you want to achieve or distribute over time. In our problem, this was \( \$5,000,000 \).
• Note the interest rate, expressed as a decimal. Here, it's \( 0.05 \) for 5%.
• Determine the number of periods, or years, the cash flow will occur. For the problem at hand, that's 20 years.

Plug these numbers into the formula and solve for \( W \) by isolating the variable. In the final step, you'll divide the desired total by the result of the previous calculation steps, enabling you to figure out how much each annual payment amount should be.

Interest Rate in Financial Mathematics

The interest rate in financial mathematics plays a critical role in determining the time value of money. It's the percentage increase of money over a period, reflecting the concept that a sum of money today is worth more than the same sum in the future because it can earn interest. In structured payment scenarios, understanding how the interest rate impacts the present and future values is key.
Interest rates can accrue positively for future calculations, indicating growth of the value over time. In our structured payment scenario, the 5% interest rate helps accumulate the same total value (\( \$5,000,000 \)) through smaller annual installments.
To see the impact of the interest rate:

• Add 1 to the interest rate in decimal form to reflect compounded growth.
• Raise this amount to the power of the negative number of periods (-20 in this example) to indicate the number of payment periods.
• Observe how the compounding diminishes the present value of each payment, making it crucial to establish the correct annual payment.

Structured Payment Plan

A structured payment plan is an arrangement where a lump sum payment is distributed over a period in several smaller installments. This type of financial arrangement offers several advantages, such as managing large amounts of money more effectively and minimizing immediate tax burdens.
In our example, the winner opts not to take the \( \$5,000,000 \) all at once but through annual payouts over 20 years. Here’s how such a plan works:

• It gives you regular, predictable payments, which can help with budgeting and planning expenses.
• The payments include calculated interest, reflecting the time value of money. So, even though you're getting smaller individual payments, the total distributed value should match the lump sum you would have received upfront after accounting for interest.
• This setup reflects how financial institutions and lotteries can structure payments to maintain the total value, often referred to as the present value, over the payment period.

By understanding structured payments, you can appreciate balancing the immediate use of funds with long-term financial health.